The Dependence of Free Energy on Pressure

A chemical engineer wants to determine the feasibility of making ethanol (C2H5OH) by reacting water with ethylene (C2H4) according to the equation 

Is this reaction spontaneous under standard conditions  

To determine the spontaneity of this reaction under standard conditions, we must determine AG° for the reaction by using the appropriate standard free energies of formation at 25°C from Appendix 4  

Although the reaction considered in Example 10.11 is spontaneous, other features of the reaction must be studied to see whether the process is feasible. For example, the chemical engineer will need to study the kinetics of the reaction to determine whether it is fast enough to be useful and, if it is not, whether a catalyst can be found to enhance the rate. In doing these studies, the engineer must remember that AG° depends on temperature  

if the process must be carried out at high temperatures to be fast enough to be feasible, AG° must be recalculated at that temperature using the AH° and AS° values for the reaction. 

To understand the pressure dependence of free energy, we need to know how pressure affects the thermodynamic functions that constitute free energy—that is, enthalpy and entropy (recall that G = H — TS). For an ideal gas enthalpy is not pressure-dependent. However, entropy does depend on pressure because of its dependence on volume. Consider one mole of an ideal gas at a given temperature. At a volume of 10.0 L, the gas has many more positions available for the molecules than if its volume is 1.0 L. The positional 

The absolute value of the free energy of a substance cannot be obtained. We use it symbolically here to show that it is the change in free energy that is really significant. 

We have shown qualitatively that the entropy and therefore the free energy of an ideal gas depend on its pressure. From a more detailed argument, which we will not consider here, one can show that 

To see how the change in free energy for a reaction depends on pressure, we will consider the ammonia synthesis reaction 

The first term in parentheses is AG° for the reaction. Thus we have 

Substituting these values into the equation gives 

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In this chapter we have seen that a system at constant temperature and pressure will proceed spontaneously in the direction that lowers its free energy. This is why reactions proceed until they reach equilibrium. The equilibrium position represents the lowest free energy value available to a particular reaction system. The free energy of a reaction system changes as the reaction proceeds, because free energy is dependent on the pressure 

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On the base of this approach thermodynamics of hydrogen absorbed outside and inside the (10,10) and the (20,20) single-wall carbon nanotubes with diameters 13.56 A and 27.13 A, respectively, was calculated. The dependencies of free energy F and thermodynamical potential H on applied pressure P and temperature T were calculated. The dependencies of content of hydrogen adsorbed on nanotubes m(P,T) surface on pressure and temperature were calculated from these data. For the first time the dependencies of m(P,T) with accounting of quantum effects and van der Waals forces were calculated. 

Therefore, we assume that energy and entropy are additive—each of them sums corresponding quantities of both phases taken as pure uniform bodies (i.e., we neglect surface energy or entropy on the phase contact). Memory is excluded because independent and dependent variables are taken in the same present instant. On the other hand, the pressure (2.109) and also temperature T (intensive quantities) are assumed to be the same in both phases (cf. discussion at the end of this Sect. 2.5). Using the deflnitions of free energies F for the whole system (2.12) and for both phases... 

Interpretation of this pressure dependence as an effective volume change from the reactants to the barrier top is generally unenlightening, however, as the microscopic origin of this behavior is more clearly understood in terms of solvation free energies and the dependence of these quantities on the solvent properties at different pressures. 

The dependence of the rate constant on pressure provides another activation parameter of mechanistic utility. From thermodynamics we have (dGldP)T = V, where V is the molar volume (partial molar volume in solutions). We define the free energy of activation by AG = G — SGr. where SGr is the sum of the molar free energies of the reactants. Thus, we obtain... 

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